Borisov A.V., Habirov S.V.
Equivalence transformations for equations of gas dynamics. Multiphase Systems. 19 (2024) 2. 44–48 (in Russian).
Equivalence transformations for equations of gas dynamics
A.V. Borisov, S.V. Habirov
Ufa University of Science and Technology, Ufa, Russia
Abstract
The object of research in this paper is the equations of gas dynamics with an arbitrary equation of state (specific internal energy as a function of
specific volume and entropy). It is required to find equivalence transformations that do not change the system of equations, but only change the
equation of state. Operators of one-parameter groups of equivalence transformations are found from the invariance criterion. The redefined
system of equations is integrated onto the coordinates of the operator. An infinite Lie algebra with two arbitrary functions is obtained.
Keywordsgroup analysis,
equivalence transformation,
equations of gas dynamics,
one-parameter group,
Lie algebra,
invariance condition
Article outline
The main task of group analysis is the group classification of equations with an arbitrary element, namely, to find arbitrary elements when the allowed group expands. In this case, an arbitrary element is determined up to equivalence transformations that do not change the form of equations, but only change an arbitrary element. For the equations of gas dynamics, the problem is solved in the work, where the equation of state (arbitrary element) It is taken as a function of pressure and density. In this case, the equivalence transformation is a three-parameter group. The most general equation of state is given by an arbitrary function depending on the specific volume and entropy.
As a result, operators of one-parameter groups of equivalence transformations for equations of gas dynamics with an equation of state in the form of an arbitrary function of specific internal energy depending on specific volume and entropy are calculated. An infinite Lie algebra with two arbitrary functions is obtained. The finite-dimensional part of this algebra is 14-dimensional.
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